def centered_figurate(k, n, show_lines=False):
"""
Show the first n centered k-gonal numbers
"""
draw.make_blank_canvas([15, 15], facecolor="lightgray")
draw.make_blank_plot(1, 1, 1, [-4, 4], [-4, 4])
draw.circle_xy(0, 0, .1)
th = np.linspace(0, 2 * np.pi, k + 1)
for i in range(1, n):
x = np.sin(th[:k]) * (3.5 * i / (n - 1))
y = np.cos(th[:k]) * (3.5 * i / (n - 1))
for p in range(k):
x_space = np.linspace(x[p % k], x[(p + 1) % k], i + 1)
y_space = np.linspace(y[p % k], y[(p + 1) % k], i + 1)
draw.circles_xy(x_space, y_space, [.1] * (i + 1))
if show_lines:
draw.closed_curve_xy(x_space,
y_space,
color="black",
zorder=-1,
lw=5)
def partition_art(n, cavas_size=[15, 15]):
draw.make_blank_canvas(cavas_size, facecolor="lightgray")
draw.make_blank_plot(1, 1, 1, [-4, 4], [-4, 4])
draw.title(f"Partitions of {n}", size=30)
parts = [i for i in partitions(n)]
xunit = 7 / n
yunit = 7 / len(parts)
y = -3.5
for p in parts:
x = -3.5
for section in p:
draw.rect_xy(x,
y,
x + section * xunit,
y + yunit,
facecolor="salmon",
edgecolor="black",
lw=3)
x += section * xunit
y += yunit
def show_primitive_pthyagorean_triples(n):
"""
Plot out the primitive integer right triangles with hypotenuse less than or equal n
"""
draw.make_blank_canvas([15, 15], facecolor="lightgray")
draw.make_blank_plot(1, 1, 1, [-.5, n], [-.5, n])
for T in primitive_pythagorean_triples():
if T[2] > n:
break
pt1 = (0, 0)
pt2 = (T[2], 0)
pt3 = (T[2], T[1])
draw.connection_p(pt1, pt2, alpha=.2)
draw.connection_p(pt2, pt3, alpha=.2)
draw.connection_p(pt3, pt1, alpha=.2)
def pascal_triangle_art(n,
circle_size=.15,
text_size=8,
circle_color="lavender",
cavas_size=15):
draw.make_blank_canvas([cavas_size, cavas_size], facecolor="lightgray")
draw.make_blank_plot(1, 1, 1, [-4, 4], [-4, 4])
draw.title("Pascal's Triangle", size=30)
for k, row in enumerate(pascal_triangle(), 0):
y = 2.5 - 4 * k / n
x_space = np.linspace(-3.7 * k / n, 3.7 * k / n, k + 1)
for x, val in zip(x_space, row):
draw.circle_xy(x, y, circle_size, fc=circle_color)
draw.text_xy(x, y, val, ha='center', va='center', size=text_size)
if k >= n:
break
def sum_of_triangular(n):
"""
Show that the sums of consecutive triangular numbers are square
"""
draw.make_blank_canvas([12, 12], facecolor="lightgray")
draw.make_blank_plot(1, 1, 1, [-4, 4], [-4, 4])
draw.title("$T_n + T_{n+1} = n^2$", size=35)
xs = np.linspace(-3, 3, n)
ys = np.linspace(-3, 3, n)
radius = 3 / (n + 2)
for m, x in enumerate(xs):
for n, y in enumerate(ys):
if m >= n:
draw.circle_xy(x, y, radius, fc="black")
else:
draw.circle_xy(x, y, radius, fc="red")
def dyck_mountains(n=2):
num_words = comb(2 * n, n) // (n + 1)
size = 1
while size * size < num_words:
size += 1
W = dyck_words(n)
draw.make_blank_canvas([15, 15], facecolor="lightgray")
for pos, word in enumerate(W, 1):
draw.make_blank_plot(size, size, pos, [-4, 4])
x = -4
y = -4
for move in word:
draw.connection_p((x, y), (x + (4 / n), y + (4 / n) * move))
x = x + (4 / n)
y = y + (4 / n) * move
def motzkin_mountains(n=2):
num_words = sum(
[comb(n, 2 * k) * c for k, c in zip(range(0, n // 2 + 1), catalan())])
size = 1
while size * size < num_words:
size += 1
W = motzkin_paths(n)
draw.make_blank_canvas([15, 15], facecolor="lightgray")
for pos, word in enumerate(W, 1):
draw.make_blank_plot(size, size, pos, [-4, 4])
x = -4
y = -3.95
for move in word:
draw.connection_p((x, y), (x + (8 / n), y + (8 / n) * move))
x = x + (8 / n)
y = y + (8 / n) * move
def pthyagorean_butterfly(hi, lo):
"""
Butterfly pattern made using integer right triangles
"""
draw.make_blank_canvas([15, 15], facecolor="lightgray")
draw.make_blank_plot(1, 1, 1, [-hi, hi], [-hi, hi])
for T in primitive_pythagorean_triples():
if T[2] > hi:
break
pt1 = (0, 0)
pt2 = (T[2], 0)
pt3 = (T[2], T[1])
draw.connection_p(pt1, pt2, alpha=.2)
draw.connection_p(pt2, pt3, alpha=.2)
draw.connection_p(pt3, pt1, alpha=.2)
pt2 = (-T[2], 0)
pt3 = (-T[2], T[1])
draw.connection_p(pt1, pt2, alpha=.2)
draw.connection_p(pt2, pt3, alpha=.2)
draw.connection_p(pt3, pt1, alpha=.2)
if T[2] <= lo:
pt2 = (T[2], 0)
pt3 = (T[2], -T[1])
draw.connection_p(pt1, pt2, alpha=.2)
draw.connection_p(pt2, pt3, alpha=.2)
draw.connection_p(pt3, pt1, alpha=.2)
pt2 = (-T[2], 0)
pt3 = (-T[2], -T[1])
draw.connection_p(pt1, pt2, alpha=.2)
draw.connection_p(pt2, pt3, alpha=.2)
draw.connection_p(pt3, pt1, alpha=.2)
def pascal_sierpinski_art(n,
circle_size=.12,
text_size=8,
circle_color="cornflowerblue",
cavas_size=15):
draw.make_blank_canvas([cavas_size, cavas_size], facecolor="lightgray")
draw.make_blank_plot(1, 1, 1, [-4, 4], [-4, 4])
draw.title("Sierpinski Triangle\nBuilt From Pascal's Triangle", size=30)
for k, row in enumerate(pascal_triangle(), 0):
y = 2.5 - 4 * k / n
x_space = np.linspace(-3.7 * k / n, 3.7 * k / n, k + 1)
for x, val in zip(x_space, row):
if val % 2 == 1:
draw.circle_xy(x, y, circle_size, fc=circle_color)
else:
draw.circle_xy(x, y, circle_size, fc="white", ec="black")
draw.text_xy(x, y, val, ha='center', va='center', size=text_size)
if k >= n:
break
def motzkin_chords_art(n, cavas_size=[15, 15], rows=10, cols=10):
draw.make_blank_canvas(cavas_size, facecolor="lightgray")
th = np.linspace(0, 2 * np.pi, n + 1)
x = np.sin(th[:n])
y = np.cos(th[:n])
P = draw.xy_to_points(x, y)
for ctr, i in enumerate(motzkin_chords(n), 1):
draw.make_blank_plot(rows, cols, ctr, [-1.2, 1.2], [-1.2, 1.2])
draw.circle_xy(0, 0, 1, fc='white', ec='black', zorder=-1)
for pair in i:
draw.connection_p(P[pair[0]], P[pair[1]], color='red', zorder=0)
draw.circles_xy(x, y, [.1] * n)